Laboratory of mathematical methods and
models in bioinformatics
Institute for Information Transmission
Problems, Russian Academy of Sciences
[email protected],
http://lab6.iitp.ru/en/
The directions below may be
of mutual interest:
I) Inferring scenarios of co-evolution of multiple
genes and regulations across the species tree.
E.g., co-evolution of a species, regulon, regulation
factor and its binding site.
II) Inferring clusters of evolutionary events
across the species tree.
E.g., several genes involved in one metabolic
pathway often undergo evolutionary perturbations at
the same areas of the species tree
To develop with tasks I-II:
1) a concept of the scenario, where each event is
assigned a particular type and the area in the
species tree;
2) an algorithm of embedding of a gene tree onto the
species tree: an algorithm of constructing the
scenario;
3) obtain a confident and representative set of gene
trees and a corresponding species tree;
4) an accurate and fast algorithm of supertree construction (also as an independent research direction)
III) Reconstructing evolution of a regulatory
region (= a certain regulation) or a gene along a
gene tree with or without defining the tree topology
and branch lengths; and inferring time slices.
An original approach is proposed in Lyubetsky,
Zhizina, Rubanov, 2008;
it can be further elaborated together
There are two fundamentally different
approaches: “inferring the events” and
“inferring the sequence – structure evolution”.
I’ll speak on the “events” (here) and then on the
“sequence – structure evolution”
(the latter is quite broad, see file Directions 2-4).
We tried to merge them but
this is a separate task
I) Co-evolution of
species, genes and regulatory elements
II) The evolution of gene
(defined by gene tree G)
along species tree S and
clustering of evolutionary events.
The separate problem of time slices here!
Example result of task I: co-evolution of
species, genes and regulatory elements
Fig. 1. Frequency profile of the 8bp long binding site and its
weakly conserved 3'-end upstream genes proA and proB widely
represented among γ-proteobacteria. The genes often form the
proBA operon. We identified a TetR family protein, an ortholog of
the NP_249058 protein from P. aeruginosa PAO1, as a
transcription factor. Sites, genes, factors and species evolve
together. The question is how?
Supertree S = species tree S
The supertree S with proBA evolutionary scenario:
S in beige color, shown inside the tubes of S
The supertree S with the factor evolutionary scenario:
S in beige color, shown inside the tubes of S
The supertree S with the site evolutionary scenario:
S in beige color, shown inside the tubes of S
Example results of task II:
In analyses of 1500 genes, 138 HGTs are
found to occur in the genus PSEUDOMONAS
being a donor of 25 and acceptor of 29 HGTs.
Other HGTs were distributed more of less
equally among other genera.
Genera: firmicutes (3), actinobacteria (1),
alpha-proteo (3), beta-proteo (2)
The upper 4 genera are Gram-positive, the rest are Gram-negative.
We see the clades of firmicutes (1-3 from the top), acinobacteria
(4), alpha-proteobacteria (5-7), beta-proteobacteria (11-12).
How to accomplish tasks I-II?
What is needed (= a working plan):
1) Definition of the evolutionary scenario
(= embedding f of G into S)?
2) Fast algorithm of constructing supertree S* from
set {Gi} and embedding f of G into S.
3) Single evolutionary event costs validation.
4) Gene tree data mining and rooting (!).
5) Clustering of evolutionary events and its
robustness against costs etc.; biological
interpretation of clusters
“Valid” definition and cs formalization of an embedding
is a fundamental task.
One well known solution does not account for
gene losses (at least as events)
and transfers (at all)
(Guigo R., Muchnik I., Smith T.F. 1996 Reconstruction
of ancient molecular phylogeny. Mol. Phyl. Evol. 6;
Mirkin and et al, …):
A known approach to reconcile the evolution of gene and
species is embedding

and its cost c(G,S)
S
G

Gene tree
Species tree
c(G,S) = 4
Inferring gene losses from embedding
“theorem”:

based on a
x
y
S
G
x

x
Gene tree
y
x
y
y
x
y
Species tree
The number of losses is a sum of one-way duplications
and gaps. Here the sum is 3 (HGTs not concerned at all)
In the context of this approach “alpha” we introduced a TEST for
putative recent HGTs: gene g is embedded into s but its
neighborhood embeds far from s

radius
1) Under this definition we revealed a long
biologically reasonable list of HGTs and drawn
some general conclusions, e.g.:
on average, one putative recent HGT decreases
the number of losses by 4.4:
lostnew  lostold  4.4  t .
But duplication counts drop only slightly.
2) We developed algorithms of finding
RECENT and ANCIENT HGTs
In our novel approach:
(“tube” is simply an edge in S)
we study embedding f of G into S, such
f ( g ) is tube d or vertex s in S,
but informally f ( g ) is a tag of the
particular event type in d or s.
We developed effective approach
to deal with such embeddings f
Scenario
 is minimal embedding f
according to functional
c( f , G, S )  cl  l ( f , Gf , S )  cd  d ( f , G, S ) 
*


 ct   t ( f , G, S )  ct   t ( f , G, S )
1) So, scenario  is a system of complex
notations of evolutionary events
in the tubes of tree S.
Our algorithm constructs scenarios and has a
cubic complexity.
2) We introduce a concept of time slices in
species tree S and developed an ad hoc
algorithm to compute time slices
Initial step of
building
f
*
Inductive steps of
building
f
*
A scenario (without HGTs) of gene evolution (defined
by gene tree G) along species tree S is minimal
mapping f of all vertices V(G) in tree G into vertices
V(S) and tubes E(S) in tree S, when the following is
true: 1) the super-root in G is mapped into the root
tube in S; each leaf g in G maps in S into leaf s, the
source of g; 2) if g1 descends from g and f(g) is a
vertex, then f(g1) < f(g), and if f(g) is a tube, then f(g1)
≤ f(g); 3) let g1 and g2 be descendants of g: if f(g) is a
vertex, then the shortest path from f(g1) into f(g2) in S
includes f(g)
A scenario (with HGTs) of gene evolution (defined by gene tree G)
along species tree S is minimal mapping f of all vertices V(G') in a
subdivision G' of G into vertices V(S) and tubes E(S) in S, when the
following is true: 1) the super-root in G' maps into the root tube in S;
each leaf g in G' maps into leaf s in S, the source of g. Let g, g1, g2 be
vertices in G'; 2) let g1 descend from g: if f(g) is a vertex, then f(g1) <
f(g), and if f(g) is a tube, then two cases apply. If g2 is another
descendant of g, then for both descendants f(gi) ≤ f(g) or: for one
descandant f(gi) ≤ f(g) and for the other f(g) ≠ f(gj) ~ f(g); here f(gi) is a
vertex of a tube and f(gj) is a tube, i,j=1,2. If g with its parent g' produce
a single descendant g1, then f(g1) ≤ f(g) ~ f(g') ≠ f(g) or f(g) ≠ f(g1) ~ f(g);
here in the first inequality f(g1) is a vertex or a tube, f(g') is a tube, and in
the second inequality f(g1) is a tube; 3) let g1 and g2 descend from g: if
f(g) is a vertex, then the shortest path from f(g1) to f(g2) in S includes
f(g); if g produces a single descendant, then f(g) is a tube.
Gene duplication is vertex g in G' with two descendants
g1 and g2, for which f(g) is a tube in S and for both
descendants f(gi) ≤ f(g), i=1,2.
Gene loss is pair <e,s>, where e is an edge in G', s – a
vertex in S with two descendants, and f(e+) < s < f(e–).
Speciation event (with respect to a given gene) is
vertex g in G', for which f(g) is a vertex in S, and each of
vertices g and f(g) produces two descendants.
Horizontal transfer with retention is vertex g in G' with
two descendants g1 and g2, for which f(g) is a tube in S,
and one of descendants gi has f(g) ≠ f(gi) ~ f(g).
Horizontal transfer without retention is vertex g in G' with
single descendant g1, for which f(g) ≠ f(g1) ~ f(g).
Constructing supertree S*
The problem of building a species supertree
given a set of gene trees {Gi} is of great applied
value. This problem is NP-hard and finding
effective solutions requires its biologically valid
reformulation.
We proposed such a reformulation and a fast
algorithmic solution to build a supertree.
Simulations and a mathematic proof demonstrate
that the algorithm is both fast and accurate
In our original approach, binary supertree S* is
sought among such trees, that have all clades
contained in fixed predefined set P of possible
clades. In the simplest case P consists of all
clades of given gene trees {Gi}.
Define V from P as basic if it can be split in two
sets from P, which can also be split in two, and so
on until singlet leaves are obtained
Supertree construction
Given set {Gi } of gene trees,
tree S(V) is built inductively:
if V is split into V1 and V2 and S(V1) and S(V2) are
already built, then they are merged on a
minimal partition V1* and V2* according to
functional
[c  l (V ,V ,V , G )  c
l
i
1
2
i
d
 d (V ,V1 ,V2 , Gi )] 
 c({Gi }, S (V1 ))  c({Gi }, S (V2 ))
The resulting tree is minimal according to
functional
 cl  l (l , Gi , S )  cd  d (l , Gi , S ) 

c( f ,{Gi }, S )   i 
 c   t  ( f , Gi , S )  c   t  ( f , Gi , S ) 
t
 t

Setting the costs in a scenario without transfers:
single duplication cost: 3,
single loss cost: 2,
single «speciation» cost in G: 0.
Later we define:
cost of HGT with retention: 11,
cost of HGT without retention: 13.
Our clustering is robust
against the cost values!
Allowing for horizontal transfers usually simplifies
scenarios. Here is a scenario of the same G and S as
before but with HGTs:
Here an edge in the gene
tree can transfer from one
tube into another within the
same slice (the problem of
slices!). Under such scenario,
no duplications are inferred
but only 1 loss and 2 HGTs
(one with and one without
retention of the donor copy)